Singularity analysis of a class of parallel robots based on Grassmann–Cayley algebra

In this paper we present a different approach for analyzing singularities of parallel robots. This approach is based on inspecting the actuators’ line dependencies by use of a matrix called the superbracket, which is similar to the Jacobian matrix and contains Plücker coordinates in its columns. Certain manipulations on this matrix, under rules based on Grassmann–Cayley algebra, enable us to obtain an algebraic statement which is then translated back into conditions between geometric entities. This method allows us to analyze the singularity in a coordinate-free manner resulting from this matrix. We demonstrate the application of this method on a class of Gough–Stewart platforms. For this class we obtained that the singular configurations occur precisely when four planes, defined by the positions of the joints, are concurrent in a point.